Problem: The sum of two numbers is $77$, and their difference is $45$. What are the two numbers?
Let $x$ be the first number, and let $y$ be the second number. The system of equations is: ${x+y = 77}$ ${x-y = 45}$ Solve for $x$ and $y$ using elimination. Add the top and bottom equations together. $ 2x = 122 $ $ x = \dfrac{122}{2} $ ${x = 61}$ Now that you know ${x = 61}$ , plug it back into $ {x+y = 77}$ to find $y$ ${(61)}{ + y = 77}$ ${y = 16}$ You can also plug ${x = 61}$ into $ {x-y = 45}$ and get the same answer for $y$ ${(61)}{ - y = 45}$ ${y = 16}$ Therefore, the larger number is $61$, and the smaller number is $16$.